To solve the problem, we need to find the area of the triangle formed by the tangent to the parabola \( y^2 = 4x \) at a point whose abscissa lies in the interval \([a^2, 4a^2]\), the ordinate, and the X-axis. We will follow these steps:
### Step-by-Step Solution:
1. **Identify the Point on the Parabola**:
The parabola is given by \( y^2 = 4x \). The points on the parabola can be expressed in parametric form as \( (t^2, 2t) \), where \( t \) is a parameter. The abscissa (x-coordinate) of the point on the parabola lies in the interval \([a^2, 4a^2]\). Therefore, we have:
\[
a^2 \leq t^2 \leq 4a^2
\]
This implies:
\[
a \leq t \leq 2a
\]
2. **Equation of the Tangent**:
The equation of the tangent to the parabola at the point \( (t^2, 2t) \) is given by:
\[
y = tx - t^2
\]
3. **Finding the Intercepts**:
- **X-intercept**: Set \( y = 0 \) in the tangent equation:
\[
0 = tx - t^2 \implies x = t
\]
- **Y-intercept**: Set \( x = 0 \) in the tangent equation:
\[
y = -t^2
\]
4. **Vertices of the Triangle**:
The vertices of the triangle formed by the tangent line, the ordinate, and the X-axis are:
- Point \( A(0, -t^2) \) (Y-intercept)
- Point \( B(t, 0) \) (X-intercept)
- Point \( C(0, 0) \) (Origin)
5. **Area of the Triangle**:
The area \( A \) of the triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is \( t \) (the X-intercept) and the height is \( -t^2 \) (the Y-intercept):
\[
A = \frac{1}{2} \times t \times (-t^2) = -\frac{1}{2} t^3
\]
Since area cannot be negative, we take the absolute value:
\[
A = \frac{1}{2} t^3
\]
6. **Maximizing the Area**:
To find the maximum area, we need to maximize \( \frac{1}{2} t^3 \) over the interval \( [a, 2a] \). The function \( t^3 \) is monotonically increasing, so we evaluate it at the endpoints:
- At \( t = 2a \):
\[
A = \frac{1}{2} (2a)^3 = \frac{1}{2} \times 8a^3 = 4a^3
\]
- At \( t = a \):
\[
A = \frac{1}{2} a^3
\]
The maximum area occurs at \( t = 2a \).
### Final Answer:
Thus, the greatest area of the triangle is:
\[
\boxed{4a^3}
\]
